What is 8. 19 divided by 4. 2 and show your work

Answer:

The correct answer is the option 3

If the lengths of all the sides of a polygon are tripled, but the angles remain the same, the area of the polygon will increase by a factor of 9. This is because the area of a polygon is directly proportional to the square of its side length. Therefore, tripling the side lengths will increase the area by a factor of 3^2, which is 9.

When the lengths of all the sides of a polygon are tripled while the angles remain the same, the new polygon will be similar to the original one but with larger sides. To determine what happens to the area of the polygon in this case, let's consider the following steps:

1. All the sides of the polygon are tripled. This means that each side's length is now 3 times its original length.

2. The angles of the polygon remain the same, so the overall shape is preserved.

3. To find the area of the new polygon, we can use the formula for the area of a similar polygon: (New area) = (scale factor)^2 * (Original area), where the scale factor is the ratio of the new side length to the original side length.

4. In this case, the scale factor is 3 (since the lengths of the sides are tripled). So, we have (New area) = (3)^2 * (Original area).

5. Therefore, the new area is 9 times the original area.

In conclusion, when the lengths of all the sides of a polygon are tripled and the angles remain the same, the area of the polygon increases by a factor of 9.

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The confidence interval:

Lower Endpoint: 6.418

Upper Endpoint: 6.782

To construct the confidence interval, we can use the formula:

CI = x ± z(σ/√n)

Where:

x = sample mean = 6.6

σ = population standard deviation = 2.3

n = sample size = 432

z = z-score for 90% confidence level = 1.645 (from the standard normal distribution table)

Plugging in the values, we get:

CI = 6.6 ± 1.645(2.3/√432)

CI = 6.6 ± 0.182

Therefore, the 90% confidence interval for the true mean number of reproductions per hour for the bacteria is:

Lower Endpoint: 6.418

Upper Endpoint: 6.782

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In hypothesis testing, the null hypothesis (H0) is a statement that there is no significant difference between two groups, or that a certain parameter is equal to a specified value. In this case, the null hypothesis is that the mean difference is 0, meaning there is no significant difference between the two groups being compared.

The alternative hypothesis (Ha), on the other hand, is a statement that contradicts the null hypothesis. It can be one-tailed (directional), indicating that the mean difference is greater than or less than 0, or two-tailed (non-directional), indicating that the mean difference is not equal to 0. So, to determine the null and alternative hypotheses in this case, we know that the null hypothesis is that the mean difference is 0. The alternative hypothesis can be either one of the following:
- Ha: The mean difference is not equal to 0 (two-tailed)
- Ha: The mean difference is greater than 0 (one-tailed)
- Ha: The mean difference is less than 0 (one-tailed)

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The number of MAD's that represents the difference of the means of each data-set is given as follows:

C. 0.2

The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations, which is also called the cardinality of the data-set.

Hence, for the first period, the mean is obtained as follows:

Mean = (3 x 0 + 4 x 1 + 5 x 2 + 2 x 3 + 1 x 4)/(3 + 4 + 5 + 2 + 1)

Mean = 1.6.

For the second period, the mean is obtained as follows:

Mean = (4 x 0 + 5 x 1 + 4 x 2 + 0 x 3 + 2 x 4)/(4 + 5 + 4 + 0 + 2)

Mean = 1.4.

The difference is then given as follows:

1.6 - 1.4 = 0.2 -> which is 0.2 MADs, as MAD = 1.

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Given a significance level (alpha) of 0.05, we can conclude that the P-value of 0.097 is greater than the alpha. Therefore, we fail to reject the null hypothesis and conclude that zip code and diet are independent of one another.

Since the P-value (0.094) is greater than the significance level of 0.05, we fail to reject the null hypothesis that zip code and diet are independent of one another. Therefore, the conclusion is that Zip code and diet are independent of one another.
We need to first calculate the degrees of freedom for the chi-square test statistic. For a contingency table with R rows and C columns, the degrees of freedom (df) is calculated as follows:
The degree of freedom of the x statistic is calculated as (number of rows - 1) times (number of columns - 1), which in this case is (4-1) times (3-1) = 6.
df = (R - 1) * (C - 1)
In this case, the table has 4 rows (zip codes) and 3 columns (diets), so:
df = (4 - 1) * (3 - 1) = 3 * 2 = 6
The test statistic (x^2) is 10.78, and the degrees of freedom is 6. To find the P-value, we need to refer to a chi-square distribution table or use statistical software. For this example, we'll round the P-value to 3 decimals.
P-value ≈ 0.097
Therefore, we need to use a chi-square distribution table with 6 degrees of freedom. Looking up the value of 10.78 in the table, we find that the P-value is approximately 0.094.
Given a significance level (alpha) of 0.05, we can conclude that the P-value of 0.097 is greater than the alpha. Therefore, we fail to reject the null hypothesis and conclude that zip code and diet are independent of one another.

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The equation of the plane tangent to the surface z = x²y + xy² + ln x + R at (1, 0, R) is z = x + y + (R - 1).

To find the equation of the plane tangent to the surface z = x²y + xy^2 + ln x + R at the point (1, 0, R), we'll need to find the partial derivatives with respect to x and y, and then use the point-slope form of the tangent plane equation. Here are the steps:

1. Find the partial derivatives of the surface function with respect to x and y:
∂z/∂x = 2xy + y² + (1/x)
∂z/∂y = x² + 2xy

2. Evaluate the partial derivatives at the given point (1, 0, R):
∂z/∂x(1, 0) = 2(1)(0) + (0)² + (1/1) = 1
∂z/∂y(1, 0) = (1)² + 2(1)(0) = 1

3. Use the point-slope form of the tangent plane equation:
z - z₀ = a(x - x₀) + b(y - y₀)

4. Substitute the point (x₀, y₀, z₀) = (1, 0, R) and the partial derivative values a = ∂z/∂x = 1, b = ∂z/∂y = 1:
z - R = 1(x - 1) + 1(y - 0)

5. Simplify the equation:
z - R = x - 1 + y

6. Rearrange the equation to the standard form:
z = x + y + (R - 1)

The equation of the plane tangent to the surface z = x²y + xy² + ln x + R at (1, 0, R) is z = x + y + (R - 1).

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The zeros of the function include the following: A. 2 and -3.

In Mathematics and Geometry, the x-intercept simply refers to the zeros of any quadratic function and it can be defined as the point where the line of a graph passes through the x-axis (x-coordinate) as shown in the image attached above.

Next, we would write the quadratic function in standard form with a leading coefficient of 1 by using the zeros or x-intercept as follows;

f(x) = (x - (-3))(x - 2)

f(x) = (x + 3)(x - 2)

f(x) = x² + 3x - 2x - 6

f(x) = x² + x - 6

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Complete Question:

What are the zeros of the following function?

a) 2 and -3

b) 2 and 3

c) 3 only

d) -3,2 and 3

Q1 = 44000 + (-0.6745) * 12166 = 36753 (rounded to the nearest whole number)

a. The distribution of X is normal with mean 44,000 and standard deviation 12,166.

b. Let Z be the standard normal variable. Then,

Z = (57239 - 44000) / 12166 = 1.0933

Using a standard normal table or calculator, we find that P(Z < 1.0933) = 0.8628. Therefore, the probability that a randomly selected acre in the forest has fewer than 57,239 ants is 0.8628.

c. Let Z1 and Z2 be the standard normal variables corresponding to 44,753 and 59,087, respectively. Then,

Z1 = (44753 - 44000) / 12166 = 0.0611

Z2 = (59087 - 44000) / 12166 = 1.2463

Using a standard normal table or calculator, we find that P(0.0611 < Z < 1.2463) = 0.3653. Therefore, the probability that a randomly selected acre has between 44,753 and 59,087 ants is 0.3653.

d. The first quartile corresponds to the cumulative probability of 0.25 in a standard normal distribution. Using a standard normal table or calculator, we find that the Z-score corresponding to a cumulative probability of 0.25 is approximately -0.6745. Therefore, the first quartile of the distribution of ants per acre in the forest is:

Q1 = 44000 + (-0.6745) * 12166 = 36753 (rounded to the nearest whole number)

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The television researcher watched the Simpsons for 3 and 1/4 hours.

To find the number of hours the researcher watched the Simpsons, we need to use the given information that Bart makes a bad decision every 1/4 of an hour. This means that in one hour (or 4/4 of an hour), Bart makes 4 bad decisions.

To find how many hours the researcher watched, we can divide the number of bad decisions by 4:

13 bad decisions ÷ 4 bad decisions per hour = 3.25 hours

Therefore, the researcher watched the Simpsons for 3 and 1/4 hours.

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We know that the volume of a cube is given by V = s^3, where s is the length of a side. Taking the derivative of both sides with respect to time, we get:

dV/dt = 3s^2 ds/dt

We are given that dV/dt = 77 cubic feet per second and V = 8 cubic feet. Therefore,

77 = 3s^2 ds/dt
ds/dt = 77/(3s^2)

We also know that the surface area of a cube is given by A = 6s^2. Taking the derivative of both sides with respect to time, we get:

dA/dt = 12s ds/dt

Substituting ds/dt from above, we get:

dA/dt = 12s (77/(3s^2))
dA/dt = 308/s

At the instant when the volume of the cube is 8 cubic feet, s = (8)^(1/3) = 2, since s is the length of a side. Therefore,

dA/dt = 308/2 = 154

So the rate of change of the surface area of the cube is 154 square feet per second.
To solve this problem, we will use the given information about the rate of change of volume and relate it to the rate of change of surface area. First, let's express the volume (V) and surface area (A) of a cube in terms of its side length (s):

1. Volume of a cube: V = s³
2. Surface area of a cube: A = 6s²

Now, differentiate both equations with respect to time (t):

1. dV/dt = 3s² ds/dt
2. dA/dt = 12s ds/dt

We are given that dV/dt = 77 cubic feet per second. We need to find dA/dt when the volume is 8 cubic feet.

From the volume equation (V = s³), we can find the side length (s) when the volume is 8 cubic feet:

8 = s³
s = 2 feet (since 2³ = 8)

Now, we can find ds/dt by plugging in the values for s and dV/dt into the first differentiated equation:

77 = 3(2²) ds/dt
77 = 12 ds/dt
ds/dt = 77/12 feet per second

Now that we have ds/dt, we can find dA/dt by plugging in the values for s and ds/dt into the second differentiated equation:

dA/dt = 12(2)(77/12)
dA/dt = 24(77/12)
dA/dt = 154 square feet per second

So, the rate of change of the surface area of the cube is approximately 154 square feet per second when the volume is 8 cubic feet.

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Answer: 240

Step-by-step explanation: i look it up and it says 240

i hope this helps

a. E[Y] = (0)(0.2) + (2)(0.5) + (4)(0.3) = 1.8 is the expected value for Y.

b. The cumulative distribution function of X+Y is P(X+Y = k) = Σ P(X=i)P(Y=k-i).

c. P(X=Y) is 0.3654.

d. Calculation is not correct. 3.6 is the correct value of E[XY].

In mathematics, a variable is defined as an alphabetic character that expresses a numerical value or number. A variable is used to represent an unknown quantity in algebraic equations.

(a) The expected value of Y can be calculated as E[Y] = (0)(0.2) + (2)(0.5) + (4)(0.3) = 1.8.

(b) To find the cumulative distribution function of X+Y, we first note that the sum of two independent random variables has a probability mass function given by the convolution of their respective probability mass functions. That is,

P(X+Y = k) = Σ P(X=i)P(Y=k-i)

where the sum is taken over all possible values of i such that both P(X=i) and P(Y=k-i) are nonzero. Using this formula, we can compute the cumulative distribution function of X+Y as:

F(x) = P(X+Y ≤ x) = Σ P(X+Y = k) for k ≤ x

     = Σ Σ P(X=i)P(Y=k-i) for k ≤ x

     = Σ P(X=i) Σ P(Y=k-i) for k ≤ x

     = Σ P(X=i) [tex]F_Y[/tex](x-i)

where [tex]F_Y[/tex](x) is the cumulative distribution function of Y. Since X has a Poisson(2) distribution, we can compute the cumulative distribution function of X+Y as:

F(x) = Σ P(X=i) F_Y(x-i)

     = Σ [tex]e^{(-2)} (2^i / i!) (0.2P(Y=x-i=0) + 0.5P(Y=x-i=2) + 0.3P(Y=x-i=4))[/tex]

where P(Y=x-i=k) is the probability mass function of Y.

(c) P(X=Y) can be calculated as:

P(X=Y) = Σ P(X=i, Y=i)

         = Σ P(X=i)P(Y=i) (since X and Y are independent)

         = Σ [tex]e^{(-2)} (2^i / i!) (0.2)(0) + (0.5)(e^(-2))(2^i / i!) + (0.3)(e^{(-2)})(2^i / i!)^2[/tex]

         = [tex]e^{(-4)} (0 + 0.5(2e^2/2) + 0.3(4e^2/4))[/tex]

         = 0.3654

(d) The student's calculation is not correct. To see why, let's first note that E[XY] can be computed as:

E[XY] = E[E[XY|X]] = E[XE[Y|X]]

where E[Y|X] is the conditional expected value of Y given X. Since X and Y are independent, we have E[Y|X] = E[Y] = 1.8. Therefore,

E[XY] = E[XE[Y|X]] = E[X(1.8)] = 2(1.8) = 3.6

So the correct value of E[XY] is 3.6, which is twice the value calculated by the student. The mistake the student made was in assuming that E[XY] is equal to the product of E[X] and E[Y]. This is only true if X and Y are uncorrelated, which is not the case here since X and Y are independent but not identically distributed.

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(a)The linear function that models the monthly cost C as a function of the distance driven d is:

C(d) = 0.25d + 260

(b) we predict that it would cost $625 per month to drive 1,500 miles.

A linear function is simple and easy to interpret, which makes it a useful model for practical purposes.

(a) Let's use the two data points to find the equation of the line that models the monthly cost as a function of the distance driven. The slope of the line is the change in cost over the change in distance, so we have:

slope = (460 - 380) / (800 - 480) = 80 / 320 = 0.25

The y-intercept is the cost when no distance is driven, so we have:

y-intercept = 380 - 0.25 * 480 = 260

(b) To predict the cost of driving 1,500 miles per month, we simply plug in d = 1500 into the linear function we found in part (a):

C(1500) = 0.25(1500) + 260 = $625

Therefore, we predict that it would cost $625 per month to drive 1,500 miles.

(c) The graph of the linear function is a straight line with slope 0.25 and y-intercept 260. The slope represents the rate of change of the cost with respect to the distance driven. In other words, for each additional mile driven, the cost increases by $0.25.

The y-intercept represents the fixed cost of driving the car, which includes expenses such as insurance and maintenance that do not depend on the distance driven.

(d) The y-intercept represents the fixed cost of driving the car, which includes expenses such as insurance and maintenance that do not depend on the distance driven.

(e) A linear function gives a suitable model in this situation because the relationship between the monthly cost and the distance driven is approximately linear over the range of distances we have data for. Additionally, a linear function is simple and easy to interpret, which makes it a useful model for practical purposes.

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The angle measure to the nearest degree of tan B = .5543 is 29°.

Given that tan B = 0 .5543, we need to find the measure to the nearest degree of tan B,

Since, we need to find the measurement of the angle, so we will use the concept of inverse of trigonometric functions,

tan B = 0 .5543

B = tan⁻¹ (0.5543)

B = 28.99 ≈ 29°

Hence, the angle measure to the nearest degree of tan B = .5543 is 29°.

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Andrés will be 34 years old and Juan will be 13 years old.

From the question, we shall make  Juan's age as J as well as Andrés' age as A.

According to the question, Juan is 21 years younger than Andrés, so we can write it as:

J = A - 21  --------Equation 1

The sum of their ages is 47 will be:

J + A = 47  ----------Equation 2

Then we substitute the sum of J from Equation 1 into Equation 2 to remove J and look for A:

(A - 21) + A = 47

2A - 21 = 47

2A = 47 + 21

2A = 68

A = 68 / 2

A = 34

Hence Andrés' age (A) is 34 years.

So we also need to substitute the value of A back into Equation 1 to know Juan's age (J):

J = A - 21

J = 34 - 21

J = 13

Hence, Juan's age (J) is 13 years.

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Juan is 21 years younger than Andrés and we know that the sum of their ages is 47. How old is each of them?

The particular solution is y = 2√(1 + x^2) - x + 2.

To solve the differential equation dy/dx = (1 + x^2)^(1/2), we can separate variables and integrate both sides:

∫1/(1 + x^2)^(1/2) dy = ∫dx

Using the substitution u = x^2 + 1, du/dx = 2x, we can simplify the integral on the left:

∫1/(1 + x^2)^(1/2) dy = ∫1/u^(1/2) * (1/2x) dy
= ∫1/u^(1/2) du
= 2√(1 + x^2)

Therefore, we have:

2√(1 + x^2) = x + C

where C is the constant of integration. To find the particular solution that satisfies y(0) = 2, we substitute x = 0 and y = 2 into the equation:

2√(1 + 0^2) = 0 + C

C = 2

So the particular solution is:

2√(1 + x^2) = x + 2

To check, we can verify that y(0) = 2 by substituting x = 0:

2√(1 + 0^2) = 0 + 2

2 = 2, which is true. Therefore, the particular solution is y = 2√(1 + x^2) - x + 2.

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We can proceed with finding the specific solution using either method mentioned above.

The order of the differential equation is 2.

Since the auxiliary equation has complex roots (±3i), we know that the general solution to the differential equation will involve sine and cosine functions.

To find the specific solution, we can use the method of undetermined coefficients or variation of parameters. However, we first need to check for any singular points or irregular singular points in the equation.

Since the coefficient of y is a polynomial in x and the coefficient of y" is also a polynomial in x, there are no singular points or irregular singular points in the equation.

Therefore, we can proceed with finding the specific solution using either method mentioned above.

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The shorter sides of heptagon as 9 feet each based on the relation, length of longer sides and total length.

Let the three shorter sides of heptagon (with seven sides) be of x feet. Hence, the remaining four sides will be of 2x feet. Now, sum of their lengths is stated thus, representing them as equation

(4 × 2x) + 3x = 99

Solving the bracket first

8x + 3x = 99

Adding the values on Left Hand Side of the equation

11x = 99

Rewriting the equation in terms of x

x = 99/11

Performing division on Right Hand Side of the equation

x = 9

Hence, the length of shorter sides is 9 feet each.

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Let X have binomial distribution b(x; 30,0.3), then P(X = 10) = 3.9 x 10^-6.

To find the probability of a specific value for X in a binomial distribution, we can use the formula:

P(X = x) = (n choose x) * p^x * (1-p)^(n-x)

where n is the number of trials, p is the probability of success in each trial, x is the number of successes we are interested in, and (n choose x) is the binomial coefficient.

In this case, we are given that X has a binomial distribution with parameters n = 30 and p = 0.3, and we want to find P(X = 10). Plugging these values into the formula, we get:

P(X = 10) = (30 choose 10) * 0.3^10 * 0.7^20

Using a calculator or software, we can calculate:

(30 choose 10) = 30,045,015

0.3^10 ≈ 0.000005

0.7^20 ≈ 0.026

Therefore,

P(X = 10) ≈ 30,045,015 * 0.000005 * 0.026 ≈ 3.9 x 10^-6

So the probability of getting exactly 10 successes in 30 trials with a success probability of 0.3 is approximately 3.9 x 10^-6.

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Max had picked 24 apples.

We have,

Let x be the number of apples Max picked.

According to the problem, if he had picked twice as many apples, he would have 24 more apples than he currently has.

This can be expressed as:

2x = x + 24

Simplifying and solving for x:

x = 24

Therefore,

Max had picked 24 apples.

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A variable without a clear operational definition can greatly impact the accuracy and reliability of statistics performed on it. To ensure accurate and meaningful results, it is important to have clear, well-defined variables in any research study.

The lack of a clear definition can lead to inconsistencies in data collection, making it difficult to accurately interpret and analyze the results.

Step 1: When a variable has no clear operational definition, researchers may measure or interpret it in different ways, leading to inconsistencies in data collection.

Step 2: These inconsistencies can affect the reliability and validity of the collected data, which in turn impacts the accuracy of the statistical analysis performed.

Step 3: As a result, the findings from such analyses may not accurately represent the true relationships or patterns within the sample or the population, leading to incorrect conclusions.

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Answer:

Area= Length × Width

Area= 45 × 15 (you get 45 as your Length because it says in the equation that the Length of the end table is 45 inches and you get 15 as the width because in the equation it says the width is 15 inches)

Answer = 45 × 15=675

The total volume of the object is 96 m^3.

A cuboid is a three dimensional shape that is formed from a rectangle. Thus its dimensions are: length, width and height.

The volume of a cuboid = length x width x height

Considering the object given in the diagram, divide it into two rectangular prisms. So that;

i. volume of rectangular prism 1 = length x width x height

                                    = 5 x 3 x 4

                                    = 60

The volume of rectangular prism 1 is 60 cubic meters.

ii. volume of rectangular prism 2 = length x width x height

                                    = 4x 3 x 3

                                    = 36

The volume of rectangular prism 2 is 36 cubic meters.

Thus,

total volume of the object = 60 + 36

                                           = 96

The total volume of the object is 96 m^3.

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Answer:

18.5

Step-by-step explanation:

Make sure the numbers are in order from least to greatest!

To define the median of the lower half, you must first divide the data set in half. Then, find the median of the first half.

17, 18, 19, 20 | 21, 24, 25, 27

To find the median, locate the number(s) in the middle of the data set.

17, 18, 19, 20

There are two numbers in the middle: 18 and 19.

When you have two medians, add the numbers, then divide the sum by 2.

[tex]18+19=37\\37/2=18.5[/tex]

The median of the lower half is 18.5.

Here's an example set of numbers:

{5, 10, 12, 15, 18, 20, 22, 22, 25, 30, 30}

Mean = sum of all numbers / total number of numbers

Mean = (5 + 10 + 12 + 15 + 18 + 20 + 22 + 22 + 25 + 30 + 30) / 11

Mean = 20

Median = the middle number when the numbers are arranged in order from smallest to largest

Median = 20 (since there are 11 numbers, the median is the 6th number, which is 20)

Mode = the most frequently occurring number in the set

Mode = 22 (since 22 appears twice in the set, which is more than any other number)

In this example, the mean and median are close together, which indicates that the data is fairly evenly distributed. However, the mode is different from the mean and median, which indicates that there is a skew in the data towards the higher end, since 22 and 30 occur twice each.

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Answer:

There is only one way to obtain a sum of 12 when rolling two standard six-sided dice, which is to get a 6 on both dice.

The probability of rolling a 6 on one die is 1/6. Therefore, the probability of rolling a 6 on both dice is:

P(D1 = 6 and D2 = 6) = P(D1 = 6) x P(D2 = 6) = 1/6 x 1/6 = 1/36

Therefore, the probability of rolling a sum of 12 with two standard six-sided dice is 1/36.

P(D1 + D2 = 12) = 1/36

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The value of S° in the given adjacent angles would be = 26.7°

Adjacent angles are those angles that are found on the same side of the plane and they share a common vertex.

The adjacent angles are different from the supplementary angles which are angles found in the same side but when measured together sums up to 180°.

The angles 41.6° and S° are two angles that share the same vertex with the sum of 68.3°

Therefore, S° which is the second part of the adjacent angles would be = 68.3+41.6 = 26.7°

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Answer:

Step-by-step explanation:

The number in your question is expressed in scientific notation, which is typically used to express numbers that are either too large or too small.  The number in scientific notation is expressed as a power of 10.  A positive exponent means the # is large whereas if the exponent is negative, then the # is small.

3.652 x 10-4 --> negative exponent, therefore, # is small.

All you need to do is to convert this number to standard notation by moving the decimal 4 places to the left.

3.652 x 10-4 = 0.0003652

Theorem 5.6.1 states that any partially ordered set of size mn has either a chain of size m or an antichain of size n.

To prove this theorem, we can use induction on m.

Base Case: When m = 1, the partially ordered set has n elements, which can be viewed as an antichain of size n or a chain of size 1.

Inductive Hypothesis: Assume that any partially ordered set of size (m-1)n has either a chain of size m-1 or an antichain of size n.

Inductive Step: Consider a partially ordered set P of size mn. We choose an element p in P, and consider the two sets:

A = {x ∈ P : x < p}

B = {x ∈ P : x > p}

Note that p cannot be compared to any element in A or B, since otherwise, we would have either a chain of length m or an antichain of length n. Therefore, p is not contained in any chain or antichain of P.

Now, we can apply the inductive hypothesis to the sets A and B. If A has a chain of size m-1, then we can add p to the end of that chain to get a chain of size m. Otherwise, A has an antichain of size n-1, and similarly, B has either a chain of size m-1 or an antichain of size n-1. If both A and B have antichains of size n-1, then we can combine them with p to get an antichain of size n.

Therefore, in all cases, we have either a chain of size m or an antichain of size n, as required. This completes the proof of Theorem 5.6.1.

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